Surface gravity

The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.

Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, which is equal to ^[1]

g = 9.80665 m/s²

In astrophysics, the surface gravity may be expressed as log g, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration and surface gravity is centimeters per second squared (cm/s²), and then taking the base-10 logarithm of the cgs value of the surface gravity. ^[2] Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665 cm/s², and then taking the base-10 logarithm ("log g") of 980.665, giving 2.992 as "log g".

The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000 g (10⁶ m/s²) whilst the neutron star's compactness gives it a surface gravity of up to 7×10¹² m/s² with typical values of order 10¹² m/s² (that is more than 10¹¹ times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.

Relationship of surface gravity to mass and radius

Surface gravity of various
Solar System bodies ^[3]

(1 g = 9.80665 m/s², the average surface gravitational acceleration on Earth)

Name	Surface gravity
Sun	28.02 g
Mercury	00.377 g
Venus	00.905 g
Earth	01 g (midlatitudes)
Moon	00.165 7 g (average)
Mars	00.379 g (midlatitudes)
Phobos	00.000 581 g
Deimos	00.000 306 g
Pallas	00.022 g (equator)
Vesta	00.025 g (equator)
Ceres	00.029 g
Jupiter	02.528 g (midlatitudes)
Io	00.183 g
Europa	00.134 g
Ganymede	00.146 g
Callisto	00.126 g
Saturn	01.065 g (midlatitudes)
Mimas	00.006 48 g
Enceladus	00.011 5 g
Tethys	00.014 9 g
Dione	00.023 7 g
Rhea	00.026 9 g
Titan	00.138 g
Iapetus	00.022 8 g
Phoebe	00.003 9–0.005 1 g
Uranus	00.886 g (equator)
Miranda	00.007 9 g
Ariel	00.025 4 g
Umbriel	00.023 g
Titania	00.037 2 g
Oberon	00.036 1 g
Neptune	01.137 g (midlatitudes)
Proteus	00.007 g
Triton	00.079 4 g
Pluto	00.063 g
Charon	00.029 4 g
Eris	00.084 g
Haumea	00.0247 g (equator)
67P-CG	00.000 017 g

In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass-produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. ^[4] For most celestial objects, the result is that the planet or star in question can be treated as a near-perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega.

The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. ^[5] Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, ^[6] and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. ^{[7] [8]} Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. ^[8]

These proportionalities may be expressed by the formula: $${\displaystyle g\propto {\frac {m}{r^{2}}}}$$ where g is the surface gravity of an object, expressed as a multiple of the Earth's, m is its mass, expressed as a multiple of the Earth's mass (5.976×10²⁴ kg) and r its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km). ^[9] For instance, Mars has a mass of 6.4185×10²³ kg = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. ^[10] The surface gravity of Mars is therefore approximately $${\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}$$ times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula $${\displaystyle g={\frac {GM}{r^{2}}}}$$ where M is the mass of the object, r is its radius, and G is the gravitational constant. If ρ = M/V denote the mean density of the object, this can also be written as $${\displaystyle g={\frac {4\pi }{3}}G\rho r}$$ so that, for fixed mean density, the surface gravity g is proportional to the radius r. Solving for mass, this equation can be written as $${\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}}$$ But density is not constant, but increases as the planet grows in size, as they are not incompressible bodies. That is why the experimental relationship between surface gravity and mass does not grow as 1/3 but as 1/2: ^[11] $${\displaystyle g=M^{1/2}}$$ here with g in times Earth's surface gravity and M in times Earth's mass. In fact, the exoplanets found fulfilling the former relationship have been found to be rocky planets. ^[11] Thus, for rocky planets, density grows with mass as ${\displaystyle \rho \propto M^{1/4}}$.

Gas giants

For gas giant planets such as Jupiter, Saturn, Uranus, and Neptune, the surface gravity is given at the 1 bar pressure level in the atmosphere. ^[12] It has been found that for giant planets with masses in the range up to 100 times Earth's mass, their gravity surface is nevertheless very similar and close to 1g, a region named the gravity plateau. ^[11]

Non-spherically symmetric objects

Most real astronomical objects are not perfectly spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects of gravitational force and centrifugal force. This causes stars and planets to be oblate, which means that their surface gravity is smaller at the equator than at the poles. This effect was exploited by Hal Clement in his SF novel Mission of Gravity , dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator.

To the extent that an object's internal distribution of mass differs from a symmetric model, the measured surface gravity may be used to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, when Roland Eötvös's torsion balance was used to prospect for oil near the city of Egbell (now Gbely, Slovakia.) ^{[13] : 1663  [14] : 223} In 1924, the torsion balance was used to locate the Nash Dome oil fields in Texas. ^{[14] : 223}

It is sometimes useful to calculate the surface gravity of simple hypothetical objects which are not found in nature. The surface gravity of infinite planes, tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into the behavior of real structures.

Black holes

In relativity, the Newtonian concept of acceleration turns out not to be clear cut. For a black hole, which must be treated relativistically, one cannot define a surface gravity as the acceleration experienced by a test body at the object's surface because there is no surface, although the event horizon is a natural alternative candidate, but this still presents a problem because the acceleration of a test body at the event horizon of a black hole turns out to be infinite in relativity. Because of this, a renormalized value is used that corresponds to the Newtonian value in the non-relativistic limit. The value used is generally the local proper acceleration (which diverges at the event horizon) multiplied by the gravitational time dilation factor (which goes to zero at the event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values of r and M.

When one talks about the surface gravity of a black hole, one is defining a notion that behaves analogously to the Newtonian surface gravity, but is not the same thing. In fact, the surface gravity of a general black hole is not well defined. However, one can define the surface gravity for a black hole whose event horizon is a Killing horizon.

The surface gravity ${\displaystyle \kappa }$ of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if ${\displaystyle k^{a}}$ is a suitably normalized Killing vector, then the surface gravity is defined by $${\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}$$ where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that ${\displaystyle k^{a}k_{a}\to -1}$ as ${\displaystyle r\to \infty }$, and so that ${\displaystyle \kappa \geq 0}$. For the Schwarzschild solution, take ${\displaystyle k^{a}}$ to be the time translation Killing vector ${\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}}$, and more generally for the Kerr–Newman solution take ${\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}$, the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where ${\displaystyle \Omega }$ is the angular velocity.

Schwarzschild solution

Since ${\displaystyle k^{a}}$ is a Killing vector ${\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}$ implies ${\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}$. In ${\displaystyle (t,r,\theta ,\varphi )}$ coordinates ${\displaystyle k^{a}=(1,0,0,0)}$. Performing a coordinate change to the advanced Eddington–Finklestein coordinates ${\textstyle v=t+r+2M\ln |r-2M|}$ causes the metric to take the form $${\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}$$

Under a general change of coordinates the Killing vector transforms as ${\displaystyle k^{v}=A_{t}^{v}k^{t}}$ giving the vectors ${\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)}$ and ${\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}$

Considering the b = ${\displaystyle v}$ entry for ${\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}$ gives the differential equation ${\textstyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}$

Therefore, the surface gravity for the Schwarzschild solution with mass ${\displaystyle M}$ is ${\displaystyle \kappa ={\frac {1}{4M}}}$ (${\displaystyle \kappa ={c^{4}}/{4GM}}$ in SI units). ^[15]

Kerr solution

The surface gravity for the uncharged, rotating black hole is, simply $${\displaystyle \kappa =g-k,}$$ where ${\textstyle g={\frac {1}{4M}}}$ is the Schwarzschild surface gravity, and ${\displaystyle k:=M\Omega _{+}^{2}}$ is the spring constant of the rotating black hole. ${\displaystyle \Omega _{+}}$ is the angular velocity at the event horizon. This expression gives a simple Hawking temperature of ${\displaystyle 2\pi T=g-k}$. ^[16]

Kerr–Newman solution

The surface gravity for the Kerr–Newman solution is $${\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}$$ where ${\displaystyle Q}$ is the electric charge, ${\displaystyle J}$ is the angular momentum, define ${\textstyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}$ to be the locations of the two horizons and ${\displaystyle a:=J/M}$.

Dynamical black holes

Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing. ^[17] Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a timelike Killing vector (field). ^[18] Several definitions have been proposed over the years by various authors, such as peeling surface gravity and Kodama surface gravity. ^[19] As of current, there is no consensus or agreement on which definition, if any, is correct. ^[20] Semiclassical results indicate that the peeling surface gravity is ill-defined for transient objects formed in finite time of a distant observer. ^[21]

References

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2. Smalley, B. (13 July 2006). "The Determination of Teff and log g for B to G stars". Keele University . Retrieved 31 May 2007.
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8. Valencia, Diana; Sasselov, Dimitar D; O'Connell, Richard J (2007). "Detailed Models of super-Earths: How well can we infer bulk properties?". The Astrophysical Journal. 665 (2): 1413–1420. arXiv: 0704.3454 . Bibcode:2007ApJ...665.1413V. doi:10.1086/519554. S2CID   15605519.
9. 2.7.4 Physical properties of the Earth, web page, accessed on line May 27, 2007.
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12. "Planetary Fact Sheet Notes".
13. Li, Xiong; Götze, Hans-Jürgen (2001). "Ellipsoid, geoid, gravity, geodesy, and geophysics". Geophysics. 66 (6): 1660–1668. Bibcode:2001Geop...66.1660L. doi:10.1190/1.1487109.
14. Prediction by Eötvös' torsion balance data in Hungary Archived 2007-11-28 at the Wayback Machine , Gyula Tóth, Periodica Polytechnica Ser. Civ. Eng.46, #2 (2002), pp. 221–229.
15. Raine, Derek J.; Thomas, Edwin George (2010). Black Holes: An Introduction (illustrated ed.). Imperial College Press. p. 44. ISBN   978-1-84816-382-9. Extract of page 44
16. Good, Michael; Yen Chin Ong (February 2015). "Are Black Holes Springlike?". Physical Review D. 91 (4): 044031. arXiv: 1412.5432 . Bibcode:2015PhRvD..91d4031G. doi:10.1103/PhysRevD.91.044031. S2CID   117749566.
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19. H. Kodama (1980). "Conserved Energy Flux for the Spherically Symmetric System and the Backreaction Problem in the Black Hole Evaporation". Progress of Theoretical Physics. 63 (4): 1217. Bibcode:1980PThPh..63.1217K. doi:10.1143/PTP.63.1217. S2CID   122827579.
20. Pielahn, Mathias; G. Kunstatter; A. B. Nielsen (November 2011). "Dynamical surface gravity in spherically symmetric black hole formation". Physical Review D. 84 (10): 104008(11). arXiv: 1103.0750 . Bibcode:2011PhRvD..84j4008P. doi:10.1103/PhysRevD.84.104008. S2CID   119015033.
21. R. B. Mann; S. Murk; D. R. Terno (2022). "Surface gravity and the information loss problem". Physical Review D. 105 (12): 124032. arXiv: 2109.13939 . Bibcode:2022PhRvD.105l4032M. doi:10.1103/PhysRevD.105.124032. S2CID   249799593.

External links

• Newtonian surface gravity
• Exploratorium – Your Weight on Other Worlds